Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 4, Issue 4, Article 74, 2003
THE SPREAD OF THE SHAPE OPERATOR AS CONFORMAL INVARIANT
BOGDAN D. SUCEAVĂ
D EPARTMENT OF M ATHEMATICS
C ALIFORNIA S TATE U NIVERSITY, F ULLERTON
CA 92834-6850, U.S.A.
bsuceava@fullerton.edu
URL: http://math.fullerton.edu/bsuceava
Received 01 May, 2003; accepted 27 September, 2003
Communicated by S.S. Dragomir
A BSTRACT. The notion the spread of a matrix was first introduced fifty years ago in algebra. In
this article, we define the spread of the shape operator by applying the same idea to submanifolds
of Riemannian manifolds. We prove that the spread of shape operator is a conformal invariant for
any submanifold in a Riemannian manifold. Then, we prove that, for a compact submanifold of
a Riemannian manifold, the spread of the shape operator is bounded above by a geometric quantity proportional to the Willmore-Chen functional. For a complete non-compact submanifold,
we establish a relationship between the spread of the shape operator and the Willmore-Chen
functional. In the last section, we obtain a necessary and sufficient condition for a surface of
rotation to have finite integral of the spread of the shape operator.
Key words and phrases: Principal curvatures, Shape operator, Extrinsic scalar curvature, Surfaces of rotation.
2000 Mathematics Subject Classification. 53B25, 53B20, 53A30.
1. I NTRODUCTION
In the classic matrix theory spread of a matrix has been defined by Mirsky in [7] and then
mentioned in various references, as for example [6]. Let A ∈ Mn (C), n ≥ 3, and let λ1 , . . . , λn
be the characteristic roots of A. The spread of A is defined to be s(A)
= maxi,j |λi −λj |. Let us
Pm,n
2
denote by ||A|| the Euclidean norm of the matrix A, i.e.: ||A|| = i,j=1 |aij |2 . We use also the
classical notation E2 for the sum of all 2-square principal subdeterminants of A. If A ∈ Mn (C)
then we have the following inequalities (see [6]):
12
2
(1.1)
s(A) ≤ 2||A||2 − |trA|2 ,
n
(1.2)
ISSN (electronic): 1443-5756
c 2003 Victoria University. All rights reserved.
060-03
s(A) ≤
√
2||A||.
2
B OGDAN D. S UCEAVĂ
If A ∈ Mn (R), then:
12
1
2
(1.3)
s(A) ≤ 2 1 −
(trA) − 4E2 (A) ,
n
with equality if and only if n − 2 of the characteristic roots of A are equal to the arithmetic
mean of the remaining two.
Consider now an isometrically immersed submanifold M n of dimension n ≥ 2 in a Riemannian manifold (M̄ n+s , ḡ). Then the Gauss and Weingarten formulae are given by
¯ X Y = ∇X Y + h(X, Y ),
∇
¯ X ξ = −Aξ X + DX ξ,
∇
for every X, Y ∈ Γ(T M ) and ξ ∈ Γ(νM ). Take a vector η ∈ νp M and consider the linear
mapping Aη : Tp M → Tp M. Let us consider the eigenvalues λ1η , . . . , λnη of Aη . We put
(1.4)
Lη (p) = sup (λiη ) − inf (λiη ).
i=1,...,n
i=1,...,n
Lη is the spread of the shape operator in the direction η. We define the spread of the shape
operator at the point p by
L(p) = sup Lη (p).
(1.5)
η∈νp M
Suppose M is a compact submanifold of M̄ .
Let us remark that when M 2 is a surface we have
L2ν (p) = (λ1ν (p) − λ2ν (p))2 = 4(|H(p)|2 − K(p)),
where ν is the normal vector at p, H is the mean curvature, and K is the Gaussian curvature.
In [1] it is proved that for a surface M 2 in E2+s the geometric quantity (|H|2 − K)dV is a
conformal invariant. As a corollary, one obtains for an orientable surface in E2+s that L2ν dV is
a conformal invariant.
Let ξn+1 , . . . , ξn+s be an orthonormal frame in the normal fibre bundle νM. Let us recall the
definition of the extrinsic scalar curvature from [2]:
s X
X
2
ext =
λin+r λjn+r .
n(n − 1) r=1 i<j
In [2] it is proved that for a submanifold M n of a Riemannian manifold (M̄ , ḡ), the geometric
quantity (|H|2 − ext)g is invariant under any conformal change of metric. If M is compact (see
also [2]), this result implies that for M , a Rn-dimensional compact submanifold of a Riemannian
n
manifold (M̄ , ḡ), the geometric quantity (|H|2 − ext) 2 dV is a conformal invariant.
Let us prove the following fact.
Proposition 1.1. Let M n be a submanifold of the Riemannian manifold (M̄ , ḡ). Then the spread
of the shape operator is a conformal invariant.
Proof. The context and the idea of the proof are similar to the one given in [3, pp. 204-205].
Let us consider ρ a nowhere vanishing positive function on M̄ . We have the conformal change
of metric in the ambient space M̄ given by
ḡ ∗ = ρ2 ḡ.
Let us denote by h and h∗ the second fundamental forms of M in (M̄ , ḡ) and (M̄ , ḡ ∗ ), respectively. Then we have (see [3]):
g(A∗ξ X, Y ) = g(Aξ X, Y ) + g(X, Y )ḡ(U, ξ),
J. Inequal. Pure and Appl. Math., 4(4) Art. 74, 2003
http://jipam.vu.edu.au/
T HE S PREAD OF THE S HAPE O PERATOR AS C ONFORMAL I NVARIANT
3
where U is the vector field defined by U = (dρ)# . Let e1 , . . . , en be the principal normal
directions of Aξ with respect to g. Then ρ−1 e1 , . . . , ρ−1 en , form an orthonormal frame of M
with respect to g ∗ , and they are the principal directions of A∗ξ . Therefore
L∗ (p) =
L∗ξ∗
sup
ξ ∗ ∈ν
p
M ;||ξ ∗ ||
∗ =1
=
sup
ξ ∗ ∈νp M ;||ξ ∗ ||∗ =1
sup (λiξ )∗
i=1,...,n
=
sup
sup
ξ∈νp M ;||ξ||=1
i=1,...,n
=
sup
sup
ξ∈νp M ;||ξ||=1
i=1,...,n
−
inf (λj )∗
j=1,...,n ξ
λjξ
j=1,...,n
λiξ
+ ḡ(U, ξ) − inf
λiξ
−
inf λjξ
j=1,...,n
+ ḡ(U, ξ)
= L(p).
This proves the proposition.
When M is a surface, both L and L2 dV are conformal invariants.
The shape discriminant of the submanifold M in M̄ w.r.t. a normal direction η was discussed
in [9]. Let Aη be the shape operator associated with an arbitrary normal vector η at p. The shape
discriminant of η is defined by
2
(1.6)
Dη = 2||Aη ||2 − (trace Aη )2 ,
n
2
1 2
n 2
where ||Aη || = (λη ) + · · · + (λη ) , at every point p ∈ M ⊂ M̄ .
The following pointwise double inequality was proved in [9]:
n
(1.7)
Dη
≤ L2η ≤ Dη ,
2
We will use this inequality later on. The proof of this fact is algebraically related to the proof of
Chen’s fundamental inequality with classical curvature invariants (see [4]). The alternate proof
of this result is presented in [10].
2. G EOMETRIC I NEQUALITIES ON C OMPACT S UBMANIFOLDS
In this section, we study the relationship between the spread of the shape operator’s spectrum
and the conformal invariant from [2]. The main result is Proposition 2.1. For its proof we need
a few preliminary steps.
Proposition 2.1. Let M n be a compact submanifold of a Riemannian manifold M̄ n+s . Then the
following inequality holds:
Z
2
Z
n2
n
n
2
.
(2.1)
LdV
(vol(M )) 2n−2 ≤ 2n(n − 1)
(|H| − ext) 2 dV
M
M
The equality holds if and only if either n = 2 or M is a totally umbilical submanifold of
dimension n ≥ 3.
Before presenting the proof, let us see what this inequality means. For any conformal diffeomorphism φ of the ambient space M̄ , the quantity
Z
2
n
LdVφ (vol(φ(M )) 2n−2
φ(M )
is bounded above by the conformal invariant geometric quantity expressed in (2.1).
First, let us prove the following.
J. Inequal. Pure and Appl. Math., 4(4) Art. 74, 2003
http://jipam.vu.edu.au/
4
B OGDAN D. S UCEAVĂ
Lemma 2.2. Let M n ⊂ M̄ n+s be a compact submanifold and p an arbitrary point in M .
Consider an orthonormal normal frame ξ1 , . . . , ξs at p and let Dα be the shape discriminant
corresponding to ξα , where α = 1, . . . , s. Then we have
s
X
1
Dα = |H|2 − ext.
2n(n − 1) α=1
(2.2)
Proof. Since
s
n
1X X i
λ
H=
n α=1 i=1 α
!
ξα ,
s
XX
2
ext =
λi λj ,
n(n − 1) α=1 i<j α α
we have
(2.3)
s
n
s X
X
1 XX i 2
2
(λα ) − 2
λiα λjα .
|H| − ext = 2
n α=1 i=1
n (n − 1) α=1 i<j
2
A direct computation yields
n
Dα =
(2.4)
2(n − 1) X i 2 4 X i j
(λα ) −
λα λα .
n
n
i<j
i=1
Summing from α = 1 to α = s in (2.4) and comparing the result with (2.3) one may get
(2.2).
From the cited result in [2] and the previous lemma, we have:
Corollary 2.3. If M is a compact submanifold in the ambient space M̄ , then
! n2
Z
s
X
Dα
dV
M
α=1
is a conformal invariant.
Let us remark that for n = 2 this is a well-known fact.
Lemma 2.4. Let M be a submanifold in the arbitrary ambient space M̄ . With the previous
notations we have
s
X
4(|H|2 − ext) ≤
L2α (p) ≤ 2n(n − 1)(|H|2 − ext)
α
at each point p ∈ M . The equalities holds if and only if p is an umbilical point.
Proof. This is a direct consequence of Lemma 2.2 and (1.7).
Proof. We may prove now Proposition 2.1. Let p be an arbitrary point of M and let η0 be a
normal direction such that L(p) = Lη0 (p). Consider the completion of η0 up to a orthonormal
normal base η0 = η1 , . . . , ηs . Then we have
(2.5)
2
L (p) =
L2η0 (p)
≤
s
X
L2α (p) ≤ 2n(n − 1)(|H|2 − ext).
α=1
J. Inequal. Pure and Appl. Math., 4(4) Art. 74, 2003
http://jipam.vu.edu.au/
T HE S PREAD OF THE S HAPE O PERATOR AS C ONFORMAL I NVARIANT
5
By applying Hölder’s inequality, one has:
Z
2 Z
2
LdV
≤
L dV (vol(M )) .
M
M
Applying Hölder’s inequality one more time yields
Z
n2
Z
n2
n−2
2
2
|H| − ext dV ≤
|H| − ext dV
(vol(M )) n .
M
M
Therefore, by using the inequality established in Lemma 2.4, we have
Z
2 Z
2
LdV
≤
L dV (vol(M ))
M
M
Z
≤ 2n(n − 1)vol(M )
|H|2 − ext dV
M
≤ 2n(n − 1) (vol(M ))
2n−2
n
Z
n
|H|2 − ext 2 dV
n2
.
M
Let us discuss when the equality case may occur. We have seen that we get an identity if
n = 2.
Now, let us assume n ≥ 3. The first inequality in (2.5) is equality at p if there exist s − 1
umbilical directions (i.e. Lα (p) = 0 for s = 2, . . . , n). The second inequality in (2.5) is
equality if and only if p is an umbilical point (see [9]). Finally, the two Hölder inequalities
are indeed equalities if and only if there exist real numbers θ and µ satisfying L(p) = θ and
|H|2 − ext = µ at every p ∈ M. The first equality conditions impose pointwise L(p) = 0,
which yields θ = µ = 0. This means that M is totally umbilical.
3. T HE N ONCOMPACT C ASE
Let M be an n-dimensional noncompact submanifold of an (n + d)-dimensional Riemannian
manifold (M̄ , g).
Proposition 3.1. Let M n ⊂ M̄ n+d be a complete noncompact
submanifold and η1 , . . . , ηd an
P
orthonormal basis of the normal bundle. Suppose that λiα λjα ≥ 0 and Lα ∈ L2 (M ). Then
Z
(|H|2 − ext)dV < ∞.
M
Proof. We use the inequality (1.7). It is sufficient to prove locally the inequality:
2
|H| − ext ≤
d
X
Di
i=1
This is true since, elementary, the following inequality holds:
2n X i j
2
(λ1α )2 + · · · + (λdα )2 −
λα λα ≤ 2[(λ1α )2 + · · · + (λdα )2 ] −
n − 1 i<j
n
( d
X
)2
(λiα )
.
i=1
This is equivalent to
n(n − 1)
d
X
i=1
(λiα )2
− 2n
2
X
i<j
J. Inequal. Pure and Appl. Math., 4(4) Art. 74, 2003
λiα λjα
2
≤ 2(n − 1)
d
X
i=1
(λiα )2 − 4(n − 1)
X
λiα λjα
i<j
http://jipam.vu.edu.au/
6
B OGDAN D. S UCEAVĂ
or
(n2 − 3n + 2)
( d
X
)
(λiα )2
+ 2(n2 − 2n + 2)
i=1
X
λiα λjα ≥ 0,
i<j
which holds by using the hypothesis and that n ≥ 2.
The inequality is the α-component of the invariant inequality we are going to prove. By
adding up d such inequalities and by considering the improper integral on M of the appropriate
functions, the conclusion follows. This is due to
X
Z
Z X
d
d Z
n
2
(|H| − ext)dV ≤
L2i dV
Di dV ≤
2
M
M
M
i=1
i=1
by the first inequality in 1.7.
In theRnext proposition we establish a relation between
integral M (|H| − ext)dV, studied in [2].
R
M
[L(p)]2 dV and the Willmore-Chen
Proposition 3.2. LetR M n ⊂ M̄ n+d be a complete noncompact orientable submanifold. If
L(p) ∈ L2 (M ), then M (|H|2 − ext)dV < ∞.
Proof. By direct computation, we have:
Z
(|H|2 − ext)dV =
(3.1)
M
1
2
n (n − 1)
Z X
d X
(λ1α − λjα )2 dV
M α=1 i<j
Z X
d X
1
≤ 2
L2 (p)dV
n (n − 1) M α=1 i<j
Z
d
=
L2 (p)dV.
2n M
Let us discuss now two examples. First, let us consider the catenoid defined by
v
v fc (u, v) = c cos u cosh , c sin u cosh , v .
c
c
Using the classical formulas for example from [8] one finds:
1
v
λ1 = −λ2 = cosh−2 .
c
c
Therefore, we have
Z ∞
Z ∞
Z ∞ t
2
e dt
−2 v
L(p)dv =
cosh
dv = 4
= 4π < ∞.
2t
c
−∞
−∞ c
−∞ e + 1
Let us consider the pseudosphere whose profile functions are given by (see, for example [5]):
c1 (v) = ae−v/a
Z vp
c2 (v) =
1 − e−2t/a dt
0
for 0 ≤ v < ∞. For simplicity, let us consider just the “upper” part of the pseudosphere. We
have
ev/a p
λ1 =
1 − e−2v/a ,
a
J. Inequal. Pure and Appl. Math., 4(4) Art. 74, 2003
http://jipam.vu.edu.au/
T HE S PREAD OF THE S HAPE O PERATOR AS C ONFORMAL I NVARIANT
v/a
λ2 = − ae
Remark that:
1−
e−2v/a
−1
.
Z
1 ∞ dy
√
√
LdV =
= ∞.
dt =
2 1
y−1
a 1 − e−2t/a
M
0
A natural question is to find a characterization for surfaces of rotation that have finite integral
of the spread of shape operator.
Consider surfaces of revolution whose profile curves are described as c(s) = (y(s), s) (see,
for example, [8]). Then we have the following.
Z
Z
∞
p
7
et/a
Proposition 3.3. Let M be a surface of rotation in Euclidean 3-space defined by
f (s, t) = (y(s) cos t, y(s) sin t, s) .
Then the integral of the spread of the shape operator on M is finite if and only if there exists
an integrable C ∞ (R) function f > 0 which satisfies the following second order differential
equation:
3
−yy 00 = 1 + (y 0 )2 ± f (s)y(1 + (y 0 )2 ) 2 .
Proof. For the proof, we use the classical formulas from [5, p. 228]. We have for λ1 = kmeridian ,
and respectively for λ2 = kparallel :
λ1 =
λ2 =
−y 00
3
[1 + (y 0 )2] 2
1
,
1 .
y[1 + (y 0 )2 ] 2
Then, the condition that the integral is finite means that there exists an integrable function f > 0
such that
Z
Z
|λ1 − λ2 |ds =
f (s)ds.
R
R
If we assume that f ∈ C ∞ , then the equality between the function under the integral holds
everywhere and a straightforward computation yields the claimed equality.
For example, for the catenoid f (s) = 0.
R EFERENCES
[1] B.Y. CHEN, An invariant of conformal mappings, Proc. Amer. Math. Soc., 40 (1973), 563–564.
[2] B.Y. CHEN, Some conformal invariants of submanifolds and their applications, Boll. Un. Mat. Ital.,
(4) 10 (1974) 380–385.
[3] B.Y. CHEN, Total Mean Curvature and Submanifolds of Finite Type, World Scientific, 1984.
[4] B.Y. CHEN, Mean curvature and shape operator of isometric immersions in real-space-forms, Glasgow Math.J., 38 (1996) 87–97.
[5] A. GRAY, Modern Differential Geometry of Curves and Surfaces with Matematica, CRC Press,
Second edition, 1998.
[6] M. MARCUS AND H. MINC, A survey of matrix theory and matrix inequalities, Prindle, Weber &
Schmidt, 1969.
[7] L. MIRSKY, The spread of a matrix, Mathematika, 3(1956), 127–130.
[8] M. SPIVAK, A Comprehensive Introduction to Differential Geometry, Volume three, Publish or
Perish, Inc., Second edition, 1979.
J. Inequal. Pure and Appl. Math., 4(4) Art. 74, 2003
http://jipam.vu.edu.au/
8
B OGDAN D. S UCEAVĂ
[9] B. SUCEAVĂ, Some theorems on austere submanifolds, Balkan. J. Geom. Appl., 2 (1997), 109–
115.
[10] B. SUCEAVĂ, Remarks on B.Y. Chen’s inequality involving classical invariants, Anal. Şti. Univ.
Al. I. Cuza Iaşi, Matematicǎ, XLV (1999), 405–412.
J. Inequal. Pure and Appl. Math., 4(4) Art. 74, 2003
http://jipam.vu.edu.au/